Average Error: 14.2 → 0.4
Time: 17.8s
Precision: 64
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r \cdot \sin b}{\left(\cos a \cdot \cos b\right) \cdot \left(\cos a \cdot \cos b\right) - \left(\left(\sin b \cdot \sin a\right) \cdot \sin b\right) \cdot \sin a} \cdot \left(\cos a \cdot \cos b + \sin b \cdot \sin a\right)\]
r \cdot \frac{\sin b}{\cos \left(a + b\right)}
↓
\frac{r \cdot \sin b}{\left(\cos a \cdot \cos b\right) \cdot \left(\cos a \cdot \cos b\right) - \left(\left(\sin b \cdot \sin a\right) \cdot \sin b\right) \cdot \sin a} \cdot \left(\cos a \cdot \cos b + \sin b \cdot \sin a\right)
double f(double r, double a, double b) {
        double r27202 = r;
        double r27203 = b;
        double r27204 = sin(r27203);
        double r27205 = a;
        double r27206 = r27205 + r27203;
        double r27207 = cos(r27206);
        double r27208 = r27204 / r27207;
        double r27209 = r27202 * r27208;
        return r27209;
}
↓
double f(double r, double a, double b) {
        double r27210 = r;
        double r27211 = b;
        double r27212 = sin(r27211);
        double r27213 = r27210 * r27212;
        double r27214 = a;
        double r27215 = cos(r27214);
        double r27216 = cos(r27211);
        double r27217 = r27215 * r27216;
        double r27218 = r27217 * r27217;
        double r27219 = sin(r27214);
        double r27220 = r27212 * r27219;
        double r27221 = r27220 * r27212;
        double r27222 = r27221 * r27219;
        double r27223 = r27218 - r27222;
        double r27224 = r27213 / r27223;
        double r27225 = r27217 + r27220;
        double r27226 = r27224 * r27225;
        return r27226;
}

Error

Bits error versus r

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 14.2

    \[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
  2. Using strategy rm

  3. Applied cos-sum0.3

    \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos a \cdot \cos b - \sin a \cdot \sin b}}\]

  4. Simplified0.3

    \[\leadsto r \cdot \frac{\sin b}{\cos a \cdot \cos b - \color{blue}{\sin b \cdot \sin a}}\]
  5. Using strategy rm

  6. Applied associate-*r/0.3

    \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos a \cdot \cos b - \sin b \cdot \sin a}}\]
  7. Using strategy rm

  8. Applied flip--0.4

    \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\frac{\left(\cos a \cdot \cos b\right) \cdot \left(\cos a \cdot \cos b\right) - \left(\sin b \cdot \sin a\right) \cdot \left(\sin b \cdot \sin a\right)}{\cos a \cdot \cos b + \sin b \cdot \sin a}}}\]

  9. Applied associate-/r/0.4

    \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\left(\cos a \cdot \cos b\right) \cdot \left(\cos a \cdot \cos b\right) - \left(\sin b \cdot \sin a\right) \cdot \left(\sin b \cdot \sin a\right)} \cdot \left(\cos a \cdot \cos b + \sin b \cdot \sin a\right)}\]
  10. Using strategy rm

  11. Applied associate-*r*0.4

    \[\leadsto \frac{r \cdot \sin b}{\left(\cos a \cdot \cos b\right) \cdot \left(\cos a \cdot \cos b\right) - \color{blue}{\left(\left(\sin b \cdot \sin a\right) \cdot \sin b\right) \cdot \sin a}} \cdot \left(\cos a \cdot \cos b + \sin b \cdot \sin a\right)\]

  12. Final simplification0.4

    \[\leadsto \frac{r \cdot \sin b}{\left(\cos a \cdot \cos b\right) \cdot \left(\cos a \cdot \cos b\right) - \left(\left(\sin b \cdot \sin a\right) \cdot \sin b\right) \cdot \sin a} \cdot \left(\cos a \cdot \cos b + \sin b \cdot \sin a\right)\]

Reproduce

herbie shell --seed 2019208 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))